Most modems are designed to be connected only to telephone lines and will not work over just a pair of wires. This is because the telephone company supplies the telephone line with a 40-50 volt DC voltage which powers part of the modem. Recall that ordinary conventional telephones are entirely powered by the voltage from the telephone company. Without such a DC voltage, the modem lacks power and can't send out data. Furthermore, the telephone company has special signals indicating a ring, line busy, etc. Conventional modems expect and respond to these signals.
One way around this is to make a simple power supply to emulate a telephone line. See Connecting two computers using their modems, without a telephone line. In most cases there are better way to connect computers together such as using network cards or just cables (null-modem) between the serial ports. Using modems has the advantage of increasing the distance as compared to a null-modem cable, since it's using a twisted pair. But it isn't nearly as fast as network cards.
A line without a source of voltage is sometimes called a "dry line" and some modems exist which are designed to work on such lines (or which can be configured to work on such lines). If you connect one of these special modems to a line with voltage on it, it may destroy the modem.
A leased line is one which is usually leased from the telephone company. The end points of the line are under the control of the user who may connect a modem at each end of the line. Leased lines may or may not have a voltage supply on them. See the mini-howto: Leased-Line which covers leased lines where there is neither voltage nor dialtone on the line. One type of leased line used two pairs of wires (one for each direction) using V.29 modulation at 9600 baud. Some brands of leased line modems are incompatible with other brands.